RAPID COMMUNICATIONS
PHYSICAL REVIEW A 82, 011803(R) (2010)
Light by light diffraction in vacuum
Daniele Tommasini and Humberto Michinel
Departamento de Fı́sica Aplicada, Universidade de Vigo, As Lagoas, s/n E-32004 Ourense, Spain
(Received 24 March 2010; published 16 July 2010)
We show that a laser beam can be diffracted by a more concentrated light pulse due to quantum vacuum effects.
We compute analytically the intensity pattern in a realistic experimental configuration, and discuss how it can be
used to measure the parameters describing photon-photon scattering in vacuum. In particular, we show that the
quantum electrodynamics prediction can be detected in a single-shot experiment at future 100-PW lasers such
as ELI or HIPER. On the other hand, if carried out at one of the present high-power facilities, such as OMEGA
EP, this proposal can lead either to the discovery of nonstandard physics or to substantial improvement in the
current limits by PVLAS collaboration on the photon-photon cross section at optical wavelengths. This example
of manipulation of light by light is simpler to realize and more sensitive than existing, alternative proposals, and
can also be used to test Born-Infeld theory or to search for axionlike or minicharged particles.
DOI: 10.1103/PhysRevA.82.011803
PACS number(s): 42.25.Fx, 12.20.Ds, 12.60.−i, 42.50.Xa
I. INTRODUCTION
The linear propagation of light in vacuum, as described by
the Maxwell equations, is a basic assumption underlying our
communication system, allowing, for example, that different
electromagnetic waves do not retain memory of their possible
crossing on the way to their reception points. However,
this superposition principle is expected to be violated by
quantum effects. In fact, quantum electrodynamics (QED)
predicts the existence of photon-photon scattering in vacuum
(PPSV) mediated by virtual charged particles running in
loop diagrams [1], although the rate is negligible in all the
experiments that have been performed up to now. On the other
hand, additional, possibly larger contributions to the process may appear in nonstandard models such as Born
Infeld theory [2–4] or in new physics scenarios involving
minicharged [5] or axionlike [6] particles. Therefore, the
search for PPSV is important not only to demonstrate a still
unconfirmed, fundamental quantum property of light, but also
to either discover or constrain these kinds of new physics.
In the last few years, there has been an increasing interest
in the quest for PPSV [7–10]. Here, we present a different
scenario to search for this phenomenon using ultra-high- power
lasers [11]. In our proposal, two almost contrapropagating
laser pulses cross each other. Because of PPSV, the more
concentrated pulse behaves like a phase object diffracting
the wider beam. The resulting intensity pattern can then be
observed on a screen and will correspond to a direct detection
of scattered photons.
II. THE EFFECTIVE LAGRANGIAN FOR
PHOTON-PHOTON SCATTERING
Following Ref. [9], we will assume that for optical
wavelengths the electromagnetic fields E and B are described
by an effective Lagrangian of the form
L = L0 + ξL L20 + 74 ξT G 2 ,
(1)
2
where L0 = 20 (E2 − c2 B ) is the Lagrangian density of the
linear theory and G = 0 c(E · B).
1050-2947/2010/82(1)/011803(4)
011803-1
In quantum electrodynamics, L would be the EulerHeisenberg effective Lagrangian [12], which coincides with
Eq. (1) with the identification ξLQED = ξTQED ≡ ξ , where
ξ=
3
8α 2h̄3
−30 m
.
6.7
×
10
45m4e c5
J
(2)
However, in Born-Infeld theory [2–4], or in models involving
a new minicharged (or milli-charged) [5] or axionlike [6]
particle, ξL and ξT will have different values, as computed
in Ref. [9].
On the other hand, the current 95% C.L. on PPSV
at optical wavelengths has been obtained by the PVLAS
Collaboration [8]. As shown in Ref. [9], it can be written
as
|7ξT − 4ξL |
m3
< 3.2 × 10−26
.
3
J
(3)
Assuming ξL = ξT ≡ ξ expt as in QED, this can be translated
into the limit ξ expt < 3.2 × 10−26 m3 /J, which is 4.6 × 103
times higher than the QED value of Eq. (2).
III. PROPOSAL FOR AN EXPERIMENT: ANALYTICAL
COMPUTATIONS
In our present proposal, illustrated in Fig. 1, a polarized
ultrahigh-power Gaussian pulse A of transverse width wA
crosses an almost contrapropagating polarized probe laser
pulse B of width wB wA . For simplicity, we assume that
the two beams have the same mean wavelength λ = 2π/k and
frequency ν = c/λ = ck/2π , although in principle they may
have different durations τA and τB . We also suppose that the
uncertainty in frequency ν is much smaller than ν, in such a
way that we can consider the pulses as being monochromatic
as a good approximation. Similarly, we assume that the
uncertainties in the components of the wave vector are much
smaller than k.
From Ref. [9], we learn that the central part of the probe B,
after crossing the pulse A, acquires a phase shift
φL,T (0) = IA (0)kτA aL,T ξL,T ,
(4)
©2010 The American Physical Society
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DANIELE TOMMASINI AND HUMBERTO MICHINEL
PHYSICAL REVIEW A 82, 011803(R) (2010)
w0 1 + (2d/kw02 )2 , and IB (0) is the peak intensity of the
wave B at the collision point, which can be related to the
total power PB = PU of the pulse B as PU = π2 wB2 IB (0).
The
√ interference term II (r) can be evaluated by multiplying
2 IU (r)ID (r) by the factor
dλ w04 − wB4 r 2
dλ
dλ
− arctan
sin
+ arctan
,
2
π wB2 w02 wU2 wD
π w02
π wB2
FIG. 1. (Color online) Sketch of the proposed experiment. An
ultraintense laser pulse A and a wider probe beam B, both moving in
a high vacuum, are focused to a region where they collide at an angle
θ close to π . The diffracted part of the probe is then observed at a
distance d on the ring screen S. (In a minimal version, a single laser
can produce both beams.)
where IA (0) is the peak intensity of the high-power beam at
the crossing point, the indices L and T refer to the two beams
having parallel or orthogonal linear polarizations, respectively,
and we have defined aL = 4 and aT = 7.
Let
AA = AA (0) exp(−r 2 /wA2 )
and
AB =
AB (0) exp(−r 2 /wB2 ) describe the dependences of the
nonvanishing components
of the two waves on the radial
2
2
coordinate r ≡ x + y orthogonal to the direction of
the motion, chosen along the z axis. The intensity of
the pulse A in the colliding region will then have the
transverse distribution IA = IA (0) exp(−2r 2 /wA2 ). As a
consequence, the space-dependent phase shift of the wave B
just after the collision with the beam A is
2r 2
(5)
φ(r) = φ(0) exp − 2 ,
wA
where we understand one of the subscripts L or T .
Because of this phase shift, the shape of the pulse B becomes
AB = AB (0) exp[−r 2 /wB2 + iφ(r)]. As discussed in Ref. [9],
φ is expected to be very small at all the facilities that will be
available in the near future. Therefore exp[iφ(r)] 1 + iφ(r)
is a very good approximation, and we obtain
r2
r2
, (6)
AB = AB (0) exp − 2 + iφ(0) exp − 2
wB
w0
where we have defined w0 ≡ (2/wA2 + 1/wB2 )−1/2 .
After the collision, the field AB propagates linearly, so that
we can just sum the free (paraxial) evolution of each term in
Eq. (6) on the screen-detector plane z = d. We then obtain
the intensity pattern I (r) = IU (r) + ID (r) + II (r), where IU
and ID correspond to the undiffracted and diffracted waves,
respectively, and II represents the interference term. The result
is
w2
2r 2
IU (r) = IB (0) B2 exp − 2 ,
wU
wU
(7)
2
2r 2
2 w0
ID (r) = IB (0)φ(0) 2 exp − 2 ,
wD
wD
where we have defined the widths
of the undiffracted and
diffracted patterns, wU ≡ wB 1 + (2d/kwB2 )2 and wD ≡
and turns out to be numerically negligible, as compared to
IU (r) + ID (r), in all the configurations that we will discuss
below.
The total power of the diffracted pulse can be obtained
by integrating Eq. (7) in the screen plane, so that PD =
π 2
w φ(0)2 IB (0), which is much smaller than PU . However,
2 0
an interesting feature of Eq. (7) is that √
the diffracted wave
is distributed in an area of width wD ∼ 2 2d/kwA wU ∼
2d/kwB (for 2d kwB2 ), so that it can be separated from
the undiffracted wave, for example, by making a hole in
the screen. Let r0 be the radius of the central region that
is eliminated from the screen. We require that the total
2
) due to
power PD (r > r0 ) = π2 w02 φ(0)2 IB (0) exp(−2r02 /wD
the diffracted wave for r > r0 is much larger than the the power
of the undiffracted wave in the same region, PU (r > r0 ) =
π 2
w I (0) exp(−2r02 /wU2 ). A safe choice might be PD (r >
2 B B
r0 ) = 100PU (r > r0 ), which implies
10wB ln
φ(0)w0
.
(8)
r0 = wD wU 2
wD − wU2
Finally, using Eqs. (4) and (7), we can compute the number
of diffracted photons that will be detected after N repetitions
of the experiment in the ring region r0 < r < R of the screen,
R being its external radius. We obtain
NDN =
8f N EA2 EB w02 −2r02 /wD2
2 2
e
− e−2R /wD (aξ )2L,T ,
4
2
π h̄c λwA wB
(9)
where f is the efficiency of the detector, and EA = PA τA and
EB = PB τB are the total energies of the two pulses.
IV. ANGULAR CONSTRAINTS AND OPTIMIZATION
OF THE SENSITIVITY
Equation (9) shows that the number of scattered photons
is proportional to the product EA2 EB of the energies of the
two laser beams. It would then be convenient to use an
ultrahigh- power pulse also for the probe B. This can be done
economically by producing both beams simultaneously, for
example, by dividing a single pulse of energy E = EA + EB
before the last focalizations. The maximum value for ND is
then obtained by taking EA = 2E/3 and EB = E/3.
The other parameters that can be adjusted in order to
maximize ND are the widths wA and wB of the two colliding
beams. The choice of wA is constrained by the requirement
that the pulse A must not spread
√ in a significant way during
> cτB λ/π . However, a more
the crossing, so that wA ∼
stringent constraint, involving also the angle θ , originates from
the condition that the center of pulse A has to remain close
to the central part of beam B during the interaction. This
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PHYSICAL REVIEW A 82, 011803(R) (2010)
implies that cτB tan(π − θ ) wA . A safe choice is then
cτB tan(π − θ ) = wA /10. On the other hand, the angle π − θ
has to be such that, away from the collision point, the
trajectories of the two beams are separated by a distance
sufficiently larger than their width. We conservatively suggest
that such a distance is six times the width ∼zλ/π wA of the
beam A at the distance z, although one should keep in mind
that smaller separations, if they turned out to be experimentally
viable, would allow for better sensitivities. For small π − θ ,
we then have π − θ 6λ/π wA , and we can solve for wA ,
wA = 60cτB λ/π.
(10)
The value of wB > wA that maximizes ND will be computed
numerically,
√ and the outer radius R will be chosen slightly
larger than 2wD ∼ 2λd/π wA , by requiring that only a small
percentage of the diffracted wave is lost.
Finally, the measurement of the number of diffracted
photons ND can be used to determine the values of the
parameters ξL and ξT . To evaluate the best possible sensitivity,
we will suppose that the background of thermal photons
and the dark count of the detectors can be made much
smaller than the signal. Although this goal may be difficult
in practice, in principle it can be achieved by cooling the
ring detector and covering it with a filter that selects a tiny
window of wavelengths around λ, and by optically isolating the
experimental area within the time of response of the detector,
which should be as small as possible (the ultrashort time of
propagation of the beams can be neglected in comparison).
Of course, the actual background should be measured by
performing the control experiments in the absence of the
beams, and with only one pulse at a time.
Under these assumptions, the best sensitivity would correspond to the detection of, say, ten diffracted photons, so that the
zero result could be excluded within three standard deviations.
The ideal minimum values of ξL and ξT that could be measured
would then be given by Eq. (9), taking NDN = 10 and all
the choices reviewed above. [In the numerical computations
that we present below, we also include a small correction
sin4 (θ/2) that appears in the expression for φ(0) as shown
in Ref. [9].]
V. SENSITIVITY AT FUTURE 100 PW LASER FACILITIES
Let us now study the possibility of performing our proposed
experiment, in its economical version discussed above, using
a 100 PW laser such as ELI [13] or HIPER [14], which are
expected to become operative in a few years. In this case, we
can use the following values: total power P = 1017 W, duration
τ = 30 fs, energy E = 3 kJ, and wavelength λ = 800 nm.
With these data, using Eq. (10), we can compute the suggested
value wA = 12 µm for the width of the spot to which the
pulse A should be focused at the collision point. Taking, for
example, d = 1 m wB2 /π λ, we find numerically that the
best choice for wB is wB 5wA = 59 µm. We then obtain
the value r0 = 2.1 cm for the central hole in the screen,
R = 4.8 cm for its outer radius, and wU = 0.43 cm and
wD = 3.1 cm for the widths of undiffracted and diffracted
waves. The focused intensities of the two beams are IA (0) =
3.1 × 1022 W/cm2 and IB (0) = 6.2 × 1020 W/cm2 , and the
angle π − θ = 0.13 rad. Even if we assume an efficiency
as small as f = 0.5, which is a realistic value today for
λ ∼ 800 nm, we obtain the result that our proposed experiment
can resolve ξL and ξT as small as ξLlimit = 2.8 × 10−30 N −1/2
m3 /J and ξTlimit = 1.6 × 10−30 N −1/2 m3 /J , values that are
well below the QED prediction even for N = 1. Therefore,
ELI and HIPER will be able to detect PPSV at the level
predicted by QED in a single-shot experiment. Two single-shot
experiments, using parallel and orthogonal polarizations of
the colliding waves respectively, would allow measurement of
both ξL and ξT .
We can restate this result in terms of the number of
diffracted photons per pulse that will be scattered in the
ring detector as predicted by QED. In the experiment with
orthogonal polarizations, this number is ∼340, which is two
orders of magnitude higher than the value obtained in the
scenario of Ref. [10], where the effect caused by a pair of
interfering, almost parallel high-power laser pulses would be
observed in the first-order diffraction maximum of the probe
beam, with a corresponding loss of intensity. This indicates
that our present proposal is by far more sensitive than that of
Ref. [10], in spite of the fact that we have applied much more
realistic assumptions on the width wA of the higher-power
pulse in order to clearly separate the beams away from the
crossing region. Moreover, the configuration of Ref. [10] is
less economical, since it requires an additional high-power
laser, and it will also present a greater experimental difficulty,
as it needs the alignment of three ultrashort pulses (two of
them having a spot radius as small as 0.8 µm ∼ λ).
Let us now compare our present proposal with that of
Ref. [9]. The main differences between the two scenarios,
both based on the crossing of two contrapropagating beams,
are the following: (1) In Ref. [9], the probe pulse could
have much smaller power; (2) in Ref. [9], the two beams
had the same width, and the effect of PPSV was observed
by measuring the phase shift of the probe by comparing
it with a third beam that was originally in phase with it.
Actually, the sensitivity calculated in Ref. [9] corresponded
to a greater power, P ∼ 1018 W, and shorter τ ∼ 10−14 s,
giving w ∼ 3 µm for I ∼ 1025 Wcm−2 . To compare that
result with the present proposal, we have to use the same
power, P ∼ 1017 W, and restate the sensitivity obtained in a
single shot in Ref. [9] as ξLlim = 2.8 × 10−8 w 2 λ/(16E) and
ξTlim = 2.8 × 10−8 w 2 λ/(28E), where E = P τ and w are the
total energy and focused width of the high-power beam. It can
be seen that a safe choice for the angle θ in that configuration
would have needed w cτ , so that even at P = 1017 W the
setup of Ref. [9] would have better used τ ∼ 10 fs instead of
30 fs. This would lead to ξLlim 1.3 × 10−29 m3 /J and ξTlim 7.2 × 10−30 m3 /J. These results would be significantly worse
than those that we have obtained above in the present proposal.
Moreover, an additional advantage of our configuration is the
fact that it can be systematically improved by increasing the
number N of crossing events in Eq. (9).
VI. SENSITIVITY AT PRESENT PETAWATT LASERS
Facilities such as OMEGA EP [15] can already provide
P = 1015 W, τ = 1 ps, and E = 1 kJ at λ = 1053 nm. From
Eq. (10), we obtain wA = 78 µm. Taking d = 1 m, we find
numerically the optimal choice wB = 5wA = 0.39 mm,
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PHYSICAL REVIEW A 82, 011803(R) (2010)
leading to r0 = 5.2 mm, R = 10 mm, wU = 0.95 mm,
wD = 6.2 mm, IA (0) = 7.0 × 1018 W/cm2 , IB (0) = 1.4 ×
1017 W/cm2 , and π − θ = 0.026 rad. Unfortunately, at
present the efficiency of photon detectors for λ ∼ 1 µm is just
f 0.1. Nevertheless, we find that our proposed experiment
can resolve ξL and ξT as small as ξLlimit = 2.1 × 10−27 N −1/2
m3 /J and ξTlimit = 1.2 × 10−27 N −1/2 m3 /J. Even for the
single-shot experiment, N = 1, this result is at least an order
of magnitude below the current limit of Eq. (3). As a result,
this experiment at present facilities could already either detect
PPSV of nonstandard origin or substantially improve the limits
on ξL and ξT . In the former case, the measurement of both ξL
and ξT can be used to discriminate between different kinds
of new physics, such as Born-Infeld theory [2–4] or models
involving minicharged [5] or axionlike [6] particles, using the
expressions for the corresponding contributions that have been
computed in Ref. [9].
Finally, we note that, in the minimal realization of our
proposal, using only one high-energy laser pulse divided into
two parts of the same time duration τA = τB = τ , the optimal
wB turns out to be proportional to wA (usually by a factor
∼5). Taking into account Eqs. (9) and (10), this implies
that the discovery potential NDN for PPSV is proportional to
f P 3 τ/λ3 . One could then study the possibility of using a
10 PW laser having a wavelength in the visible window, in
order to compensate in part the lower power (as compared
with the 100 PW case discussed above) with the higher
f and λ−3 factors. In this case, assuming τ ∼ 30 fm, our
computations show that PPSV at the QED rate can be detected
by accumulating a number of repetitions N ∼ 10.
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ACKNOWLEDGMENTS
We thank D. Novoa and F. Tommasini for useful discussions
and help. This work was supported by the Government of
Spain (Project No. FIS2008-01001) and the University of Vigo
(Project No. 08VIA09).
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